Optimal. Leaf size=28 \[ \frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x)) \]
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Rubi [A] time = 0.0255663, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4335, 3443, 2635, 8} \[ \frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4335
Rule 3443
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos (x) \cos (\cos (x)) \sin (x) \sin (\cos (x)) \, dx &=-\operatorname{Subst}(\int x \cos (x) \sin (x) \, dx,x,\cos (x))\\ &=-\frac{1}{2} \cos (x) \sin ^2(\cos (x))+\frac{1}{2} \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x))-\frac{1}{2} \cos (x) \sin ^2(\cos (x))+\frac{1}{4} \operatorname{Subst}(\int 1 \, dx,x,\cos (x))\\ &=\frac{\cos (x)}{4}-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x))-\frac{1}{2} \cos (x) \sin ^2(\cos (x))\\ \end{align*}
Mathematica [A] time = 1.51206, size = 21, normalized size = 0.75 \[ \frac{1}{4} \cos (x) \cos (2 \cos (x))-\frac{1}{8} \sin (2 \cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 23, normalized size = 0.8 \begin{align*}{\frac{ \left ( \cos \left ( \cos \left ( x \right ) \right ) \right ) ^{2}\cos \left ( x \right ) }{2}}-{\frac{\cos \left ( \cos \left ( x \right ) \right ) \sin \left ( \cos \left ( x \right ) \right ) }{4}}-{\frac{\cos \left ( x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96901, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{4} \, \cos \left (x\right ) \cos \left (2 \, \cos \left (x\right )\right ) - \frac{1}{8} \, \sin \left (2 \, \cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05015, size = 219, normalized size = 7.82 \begin{align*} \frac{1}{2} \, \cos \left (x\right ) \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{2} + \frac{1}{4} \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) - \frac{1}{4} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.89546, size = 34, normalized size = 1.21 \begin{align*} - \frac{\sin ^{2}{\left (\cos{\left (x \right )} \right )} \cos{\left (x \right )}}{4} - \frac{\sin{\left (\cos{\left (x \right )} \right )} \cos{\left (\cos{\left (x \right )} \right )}}{4} + \frac{\cos{\left (x \right )} \cos ^{2}{\left (\cos{\left (x \right )} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08866, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{4} \, \cos \left (x\right ) \cos \left (2 \, \cos \left (x\right )\right ) - \frac{1}{8} \, \sin \left (2 \, \cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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